A structure theorem for sets of small popular doubling, revisited

TL;DR

This paper establishes a stability result for sets with small popular doubling in cyclic groups, showing such sets are close to arithmetic progressions, thus extending Vosper's theorem.

Contribution

It provides a new stability theorem for sets with small popular doubling in cyclic groups, generalizing Vosper's theorem.

Findings

Sets with small popular doubling are close to arithmetic progressions.

The stability result applies to large prime cyclic groups.

It extends classical additive combinatorics results to a broader context.

Abstract

We prove that every set $A\subset\mathbb{Z}/p\mathbb{Z}$ with $\mathbb{E}_x\min(1_A*1_A(x),t)\le(2+\delta)t\mathbb{E}_x 1_A(a)$ is very close to an arithmetic progression. Here $p$ stands for a large prime and $\delta,t$ are small real numbers. This shows that the Vosper theorem is stable in the case of a single set.